Question

Suppose you have two coins which appear identical in your pocket. You know that, one is fair and one is $2$ headed. If you take one out, toss it and get a head, what is the probability that it was a fair coin?

• A. $\frac{1}{4}$
• B. $\frac{1}{5}$
• C. $\frac{1}{3}$
• D. $\frac{1}{7}$

Answer 1

Answer: (C)

Let $E_{1}$ be the event that fair coin is drawn, $E_{2}$ be the event that $2$ headed coin is drawn, and $E$ be the event that tossed coin get a head.
$\therefore P\left(E_{1}\right)=1 /2, P\left(E_{2}\right)=1/ 2, P\left(E |E_{1}\right)= 1/ 2$
and $P\left(E| E_{2}\right)=1$
Now, $P\left(E_{1} |E\right)=\frac{P\left(E_{1}\right)\cdot P\left(E| E_{1}\right)}{P\left(E_{1}\right)\cdot P\left(E| E_{1}\right)+P\left(E_{2}\right)\cdot P\left(E| E_{2}\right)}$
$=\frac{\frac{1}{2}\cdot\frac{1}{2}}{\frac{1}{2}\cdot\frac{1}{2}+\frac{1}{2}\cdot1}$
$=\frac{\frac{1}{4}}{\frac{1}{4}+\frac{1}{2}}$
$=\frac{\frac{1}{4}}{\frac{3}{4}}=\frac{1}{3}$